Put simply, renting a title larger than the remaining capacity on your device forces iOS to use a hitherto undisclosed clean-up routine, thereby freeing-up space. Even if the download is way too big, it’ll still try to free-up space.
I wish iOS was more clear about what was being deleted during this process. I assume it’s just unnecessary cache files, but I haven’t been able to find a source that explains exactly what the process is doing.
I will add, though, that this trick works when trying to download a previously purchased movie, too. As long as the file is larger than the device’s available free space, it will run the cleanup process. I tried it out on my fiancée’s 16GB iPhone 5s and was able to free up about 600MB of storage.
(Via The Loop.)